Hello, I'm trying to find the differential equation of this circuit

but I got an Integro differential equation, I don't know what's next because the current function is on terms of tao and t at the same time

 

You could take derivative with t of both sides and then there is only one variable t.

 

Ok I did it and it looks like that

But I have one more quesion, why when I am finding the differential equation, I said that in this case

but when I apply laplace for example to a derivative, I have to replace the real values of the initial conditions derivative

 

The first case Vc(0) is a constant while the second case it is the value of capacitor derivative at the time t equals to 0 (first you derivative Vc and get dVc/dt then you evaluate it at t=0).

 

mmmmm..... It all makes sense now, thank you so much